How about this:
Some pure blood Klingons are allowed to pursue non-military careers if they so choose or if they are too weak/cowardly to cut it as a warrior. They aren't sick and weak enough to be put to death, but they are pathetic Klingons that can only find work as nannies, cooks, etc...
Some Klingons choose to also diversify into other fields IN ADDITION to military service, such as Kang's wife.
Meanwhile, the majority of the menial non-military professions are left to the conquered species. These pathetic losers are not even allowed to leave their home planet or even board a ship with Warp drive. They are allowed to continue to exist solely to provide the Empire with much needed non-military infrastructure.